Why can't I make a vertical loop cut on this object?
To answer my own question. There were hidden doubles in the object. Even though the original post says that this was already checked, I removed doubles again and that time it worked.

In edit mode I deselected, then reselected all and used the remove doubles tool.
A, W, Remove Doubles.

Making sure to set my Merge Distance greater than 0. After that, I was able to add a vertical loop cut.

The other SE questions I used to help fix this were:


why does loop-cut-and-slide ignore some faces but not others?
Loop-cut won't go all the way around
Loop cut interrupted